Download Ottimizzazione delle emissioni di Titoli di Stato

Stochastic models for interest rates
in the Optimization of Public Debt
Davide Vergni
Istituto per le applicazioni del Calcolo “Mauro Picone”
Consiglio Nazionale delle Ricerche
Viale del Policlinico, 137 – 00161 Roma – Italy
http://www.iac.cnr.it/
E-mail: [email protected]
Collaboration CNR – Ministry of Economy and Finance
Istituto Applicazioni
del Calcolo:
Massimo Bernaschi
Alba Orlando
Marco Papi
Benedetto Piccoli
Davide Vergni
Alessandra Caretta
Paola Fabbri
Davide Iacovoni
Francesco Natale
Stefano Scalera
Antonella Valletta
What is the Public Debt?
Public Debt
The compound of the
yearly budget deficit
in the history
DEFICIT:
• Primary Budget Surplus: is the difference between revenues (mostly taxes)
and expenditures (mostly salaries). It can be influenced by political orientation:
social expenses, investment, selling state's property
•Interest over the Debt: expenses for the passive interest on the past debt. It
depends on the debt composition and can be modified by optimizing the debt
composition
Public Debt Management
The Growth and Stability Pact (GSP), subscribed by the
countries of the European Union (EU) in Maastricht,
defines sound and disciplined public finances as an
essential condition for strong and sustainable growth
with improved employment creation
The rules of the pact require that
The budget deficit has to be below 3% of Gross Domestic Product
The total Debt has to be less than 60% of the GDP
Gross Domestic Product: the total output of the economy (PIL)
Now the rule are less severe, because they take into account the economic cycle
Public Debt Management:
Italian situation
Deficit/GDP
Debt/GDP
2003
2,4
106,2
2004
2,9
105,9
2005
4,3
106,6
1250 billion Euros: Total amount of Italian government stock
277 billion Euros: Bonds expiring in next year
This is a very difficult situation. The only lucky fact is that the interest rate are low.
With this mass of debt the use of an optimization strategy that reduces only few
percentual point in the new issuance, lead to a remarkable money savings
A reduction of the 0.4% on the new issuance
leads to over than 1 billion euros of money savings
Public debt composition
BOT, CTZ Zero Coupon Bond
3, 6, 12 and 24 months maturity
BTP Fixed Rate Coupon Bond
3, 5, 10, 15 and 30 years maturity
CCT Floating Rate Coupon Bond
year maturity
7
BTP €i Floating Capital Coupon Bond
is similar to a BTP but its capital is
linked to the european inflation growth
The Italian Public Debt are payed mostly selling different
securities (nearly 82% of the total debt). The Italian Treasury
regularly issued five different securities:
BOT, CTZ, BTP, BTP €i and CCT.
The expenses for interest payments on Public Debt are
about 15% of the Italian Budget Deficit
Interest Rate
Is the measure, in percentage terms (interests) of the money
due by the state in one year to investors that lend money.
Yearly interest rate
Each Bond has its own interest
rate that determines the
corresponding price.
Usually, for long-term loan, the
interest rate is high.
3, 6, 12, 24, 60, 120, 180, 360
INFLATION
[issuance price, coupon]
Interest Rates Evolution
Historical term structure
How to manage Public Debt
We can manage public debt just acting on the
debt composition in terms of issued securities
IAC and
Ministry of Economy
Project
“Analisi dei problemi inerenti
alla gestione del debito
pubblico interno ed al
funzionamento dei mercati”.
Debt Management (portfolio composition) can be
seen as a constraint optimization problem
Fixing a time-window (typically 5 years) what is the
optimal debt composition which minimize the debt fulfilling
in the meantime all the istitutional and market constraints?
Optimization Structure
Stochastic Components
The most important stochastic elements of the problem are
• Primary Budget Surplus: linked to economic policy and
macroeconomic factors. It is difficult to modelize.
•Evolution of the interest rates: modeled by using of
stochastic differential equations like:
drt = μ(rt, t)+ σ (rt ,t) dBt
dft(T) = (t, T, )+ σ (t, T, ) dBt
A model for the evolution of short term rates corresponds to a
specific functional form for μ(rt, t) and σ (rt ,t). A model for the
term structure evolution corresponds to a specific functional
form for (t, T, ) and σ (t, T, )
Our model for interest rates
All rates are strongly correlated to the official discounted rate
determines by the European Central Bank (ECB).
Rates
decomposition
Therefore we can think that each rate could
be decomposed in a term proportional to
ECB and in a term ortoghonal to the ECB
Comparison interpolated ECB and rates (1)
Comparison interpolated ECB and rates (2)
Decomposition Example
First model of fluctuations - PCA
For the generation of orthogonal fluctuation
we considered a simple multivariate brownian motion
We do not use the correlated components of the stochastic terms
where Z are a nine component vector
of gaussian independent increments
but we just use three principal components of the
random noise which give 98% of the total variance
where z are a nine compoment vector of gaussian independent
increments with only the first three component different from 0
U is the diagonalization matrix for the square root of the covariance matrix, ,
and D is the diagonal matrix associated to 
Second model of fluctuations - CIR
•
•
•
Another possibility for the generation of orthogonal fluctuation is by the use of a
multivariate extension of the classical model for the short term rate by
Cox-Ingersoll-Ross (CIR-1985):
are constant verifying the condition
The settings of the model parameters is by the maximum likelihood applied to
the discrete evolution equation
Validation for the term structure
Our goal is not to forecast rates evolution, but to
generate "reasonable" scenario of rates evolution
The term structure of
interest rates could be
very different from the
historical ones
The cross-correlation of
interest rates could be
very different from the
historical ones
We control the growth and the
convexity of the generated
term structure
We control the simulated
cross correlation
Term structure example using PCA
Term structure example using CIR
Macroeconomical model
Basic Model:
ECB - Inflation
It is a completely interacting model
• the inflation modifies the monetary policy of the ECB,
• the ECB policy, on the other hand, modify the inflation
The goal is to capture the link between the inflation and the monetary policy
adopted by the ECB. Moreover we are also interested in understanding how the
intervention of the ECB reflects on the interest rates evolution in the euro area
The principal economic ingredients are:
The goal of the ecb is
to maintain the
inflation around 2%
The real Short interest
rate has to be positive
Macroeconomic variables
ECB official discount rate
Harmonized Index of Consumer Prices (HICP) ex tobacco
Annual Inflation Rate
Euristic model
The inflation evolves according to the rule
 is distributed as the historical absolute value of the inflation increments
s could be 1 or -1 according to a certain probability
The ECB rate evolves according to the rule
where
Each change of the ECB rate acts on the probability of s
Non linear model
We use coupled maps with stochastic element
At difference with the previous model now  is a random variable:
Where
and
are binomial random variables whose value can be 0 or 1,
with a probability that depends on the value of .
K and are constant values obtained by the calibration of the model,
f is a non linear function and z is a gaussian random variable
Building a complete model
Macroeconomic Factors
Official discount rate, Inflation
Microeconomic Factors
Primary Budget Surplus,
Interest rates
Gross Domestic Product
macro-micro economic model
A total interacting model
involving all the macro
and microeconomic factors
Building a complete model
Economic Cycle
Macroeconomic
Interest
Variable
Factor
Rates
A hierarchical model: each component involves
homogeneous quantities, using variables of higher level as
quasi-parameters.
The economic cycle variable is a non-observable quantity
Present State of the Project
• The software prototype is complete and running at the Ministry of
Economy
• All components have been validated on real data
• At present the scenario generator implement two different ecbinflation model and two different interest rates model.
Open problems
• Improve the interest rate models.
• Build a macroeconomic model
• Improve the cost-risk analysis